Lewis equation for beam strength of a gear tooth

Lewis (1892) treated a gear tooth as a cantilever beam loaded at its tip by the full tangential load, and found its load-carrying strength in bending.

Assumptions

• The full load $W_t$ acts at the tip of one tooth only.
• The load is uniformly distributed across the face width $b$.
• The radial component (compression) and friction are neglected.
• The effect of stress concentration at the root is neglected.
• The tooth is treated as a parabola (a beam of uniform strength), and the section of maximum stress is where the parabola touches the tooth profile.

         Wt
          |
          v
        __|__        <- tip
       /  |  \
      /   | h \  parabola
     /   _|_   \
    |<--- t --->|   critical section

Derivation

Let $t$ = tooth thickness at the critical section, $h$ = height of the load above that section, $b$ = face width.

Bending moment $M = W_t\,h$. Section modulus $Z = \dfrac{b t^2}{6}$.

$$\sigma_b = \frac{M}{Z} = \frac{6\,W_t\,h}{b\,t^2}$$

So the permissible tangential load is

$$W_t = \frac{\sigma_b\, b\, t^2}{6h}$$

The critical section is where the inscribed parabola touches the tooth profile. From the similar triangles of the parabola, $t^2 = 4hx$, where $x$ is a length fixed by the tooth profile. Substituting,

$$W_t = \frac{\sigma_b\, b \cdot 4hx}{6h} = \sigma_b\, b\,\frac{2x}{3}$$

Multiplying and dividing by the circular pitch $p_c = \pi m$:

$$W_t = \sigma_b\, b\, \pi m\left(\frac{2x}{3 p_c}\right)$$

The dimensionless term in brackets is the Lewis form factor $y$:

$$\boxed{W_t = \sigma_b\, b\, \pi\, m\, y}$$

where $\sigma_b$ is the allowable bending stress ($\sigma_o$ for static conditions, or $\sigma_o C_v$ with the velocity factor). Writing $Y = \pi y$ gives the other form, $W_t = \sigma_b b m Y$.

Form factor

$y$ depends on the tooth profile and the number of teeth $z$; it increases with $z$:

• 20 degree full depth: $y = 0.154 - \dfrac{0.912}{z}$
• 14.5 degree composite or full depth: $y = 0.124 - \dfrac{0.684}{z}$

For a given pair of gears, the design is based on the weaker of the pinion and the gear (the one with the smaller $\sigma_b y$). The pinion has fewer teeth, and so a smaller $y$, but it is often made of a stronger material.